In this concluding post on the theories of exponential and logarithmic functions we will present the most intuitive and obvious approach to define the expression $a^{b}$ directly without going to the number $e$ and the function $\log x$. This approach stems from the fact that an irrational number can be approximated by rational numbers and we can find as good approximations as we want. The idea is that if $b$ is irrational and $a > 0$ then we have many rational approximations $b', b'', \ldots $ to $b$ and the numbers $a^{b'}, a^{b''}, \ldots$ would be the approximations to the number $a^{b}$ being defined. Inherent in such a procedure is the belief that we can find as good approximations to $a^{b}$ as we want by choosing sufficiently good rational approximations to $b$. Thus we can see that the numbers $$2^{1}, 2^{1.4}, 2^{1.41}, 2^{1.414}, 2^{1.4142}, \ldots$$ are approximations to the number $2^{\sqrt{2}}$.

Exponential Function as a Limit

In the last post we developed the theory of exponential and logarithmic function using the standard approach of defining logarithm as an integral. In this post we will examine various alternative approaches to develop a coherent theory of these functions. We will start with the most common definition of $\exp(x)$ as the limit of a specific sequence. For users of MSE this is the approach outlined in this answer on MSE.

In the past few months I saw a lot of questions on MSE regarding exponential and logarithmic functions. Most students were more used to the idea of defining $e$ by $\lim\limits_{n \to \infty}\left(1 + \dfrac{1}{n}\right)^{n}$ and then defining the exponential function as $e^{x}$. I tried to answer some of these questions and based on the suggestion of a user, I am trying to consolidate my answers into a series of posts here. One thing which I must mention here is that most students do have an intuitive idea of the exponential and logarithmic functions but many lack a sound theoretical foundation. In this series of posts I will provide multiple approaches to develop a theory of exponential and logarithmic functions. We will restrict ourselves to real variables only.